Extremal inverse eigenvalue problems for matrices with a ... · To construct matrices of a certain...

Extremal inverse eigenvalue problems for

matrices with a prescribed graph

Debashish Sharma1 Bhaba Kumar Sarma2

1Department of MathematicsGurucharan College, Silchar

2Department of MathematicsIndian Institute of Technology Guwahati

1/29 D.Sharma, B.K.Sarma

Introduction

Inverse Eigenvalue Problem

The problem of reconstruction of specially structured matrices from aprescribed set of eigen data is known as an inverse eigenvalue problem,in short IEP.

Introduction

Objective of IEP

To construct matrices of a certain pre-defined structure, satisfying thegiven restrictions on eigenvalues and eigenvectors of the desired matricesor their submatrices.

Introduction

Objective of IEP

To construct matrices of a certain pre-defined structure, satisfying thegiven restrictions on eigenvalues and eigenvectors of the desired matricesor their submatrices.

Remark

The same eigen data may give rise to a completely different IEP if thestructure of the desired matrix is changed. In the same way, a slightchange in the eigen data may give rise to a completely different IEPeven though the structure of the required matrix is kept same.

Introduction Preliminaries

Matrices of a graph and Graph of a matrix (Hogben [1])

Let G be an undirected simple graph with n vertices v1, v2, . . . , vn andA = (aij) be an n× n symmetric matrix which is constructed such thatfor i 6= j, aij 6= 0 if vivj is an edge and aij = 0 if vivj is not an edge,then A is called a matrix of the graph G and G is called the graph of the

matrix A.

Matrices of a graph and Graph of a matrix (Hogben [1])

Let G be an undirected simple graph with n vertices v1, v2, . . . , vn andA = (aij) be an n× n symmetric matrix which is constructed such thatfor i 6= j, aij 6= 0 if vivj is an edge and aij = 0 if vivj is not an edge,then A is called a matrix of the graph G and G is called the graph of the

matrix A.

There is no restriction on the diagonal elements.

A given graph has infinite number of matrices associated with itbut a given matrix has a unique graph.

The set of all n× n symmetric matrices whose graph is G isdenoted by S(G).

A matrix whose graph is a tree is called an acyclic matrix.

Illustration

Graph on 6 vertices

Illustration

Graph on 6 vertices

Illustration

Graph on 6 vertices

v1 v2 v3 v4 v5 v6

Illustration

Graph on 6 vertices

v1 v2 v3 v4 v5 v6

Illustration

Graph on 6 vertices

v1 v2 v3 v4 v5 v6

A matrix of this graph

Illustration

Graph on 6 vertices

v1 v2 v3 v4 v5 v6

a11 a12

Illustration

Graph on 6 vertices

v1 v2 v3 v4 v5 v6

a11 a12 0

Illustration

Graph on 6 vertices

v1 v2 v3 v4 v5 v6

a11 a12 0 0 0 0

Illustration

Graph on 6 vertices

v1 v2 v3 v4 v5 v6

a11 a12 0 0 0 0

a12 a22 a23 a24 0 0

Illustration

Graph on 6 vertices

v1 v2 v3 v4 v5 v6

a11 a12 0 0 0 0

a12 a22 a23 a24 0 0

0 a23 a33 0 0 0

0 a24 0 a44 0 0

0 0 0 0 a55 a56

0 0 0 0 a56 a66

Here a12, a23, a24, a56 6= 0.

Problem studied

Extremal IEP

Given a graph G on n vertices, 2n − 1 real numbers αj , j = 1, 2, . . . , nand βj , j = 1, 2, 3, . . . , n with α1 = β1, find a matrix A ∈ S(G) suchthat for each j = 1, 2, . . . , n, αj and βj are respectively the smallest andlargest eigenvalues of Aj , the j × j leading principal submatrix of A.

Problem studied

Extremal IEP

Given a graph G on n vertices, 2n − 1 real numbers αj , j = 1, 2, . . . , nand βj , j = 1, 2, 3, . . . , n with α1 = β1, find a matrix A ∈ S(G) suchthat for each j = 1, 2, . . . , n, αj and βj are respectively the smallest andlargest eigenvalues of Aj , the j × j leading principal submatrix of A.

This problem was first studied by J. Peng et. al. [2] (2006) and thenby H. Pickman et. al. [3, 4] (2007,2009) for the construction of arrowmatrices and doubly arrow matrices.

Motivated by this, recently several authors studied the problem of con-structing matrices whose graphs are certain types of trees, namely,brooms (D. Sharma and M. Sen [5]), dense centipedes (D. Sharmaand M. Sen [6]), generalized stars (M. Heydari et. al. [7]), bananatrees (M.B. Zarch et. al. [8]), double-starlike trees or double comets(M.B. Zarch and S.A.S. Fazeli [9]).

Problem studied

Special trees for which extremal IEP has been studied

Problem studied

Broom Dense Centipede

Problem studied

Generalized star

Problem studied

Generalized star Banana Tree

Problem studied

Generalized star Banana Tree Double comet

Problem studied

Given a tree T on n vertices and 2n− 1 real numbers αj , βj , 1 ≤ j ≤ n,with the convention α1 = β1, find a matrix A ∈ S(T ) such that αj andβj are respectively the smallest and the largest eigenvalues of Aj .

Solving the extremal IEP for an arbitrary tree

The solutions obtained for special trees relied upon suitable ways oflabelling the vertices of the trees so as to express the characteristicpolynomials of the corresponding matrices and their leading principalsubmatrices in terms of simple recurrence relations.

Problem studied

Scheme of labelling

1 Start with any vertex of the unlabeled tree T0 and label it as 1.

2 Select a vertex in T0 adjacent to 1 and label it as 2.

3 In each subsequent step, label a vertex that is adjacent to one ofthe vertices that have already been labeled, maintaining the serialnumber of the new labels in the natural order.

4 Continue this process till all the vertices are labeled from 1 to n.

Problem studied

Scheme of labelling

1 Start with any vertex of the unlabeled tree T0 and label it as 1.

2 Select a vertex in T0 adjacent to 1 and label it as 2.

3 In each subsequent step, label a vertex that is adjacent to one ofthe vertices that have already been labeled, maintaining the serialnumber of the new labels in the natural order.

4 Continue this process till all the vertices are labeled from 1 to n.

Remark

With this scheme of labelling, the labeled tree T thus obtained has theproperty that for each j, the subgraph induced by {1, 2, . . . , j} is con-nected i.e. also a tree.

Thus, for any matrix A in S(T ), graph of each j × j leading principalsubmatrix Aj of A is also a tree. We call such a matrix as highly acyclicmatrix.

Problem studied

Illustration

Problem studied

Illustration

Problem studied

Illustration

Problem studied

Illustration

Problem studied

Illustration

Problem studied

Illustration

Problem studied

Illustration

Problem studied

Illustration

Problem studied

Illustration

11 1213 14 15

Figure: Scheme of labelling

Problem studied

Lemma 1

For 1 ≤ j ≤ n, the vertex j is pendent in the subtree Tj of T inducedby {1, 2, . . . , j}.

Problem studied

Lemma 1

For 1 ≤ j ≤ n, the vertex j is pendent in the subtree Tj of T inducedby {1, 2, . . . , j}.

This allows us to define a function v : {2, 3, . . . , n} → {1, 2, . . . , n− 1}given by v(j) = the unique vertex among 1, 2, . . . , j − 1 that is adjacentto j. Thus, for each j = 1, 2, . . . , n, the only non-zero off-diagonal entryin the jth column of the j × j leading principal submatrix Aj of anymatrix A in S(T ) is in the v(j)th row and the only non-zerooff-diagonal entry in the jth row of Aj is in the v(j)th column.

Problem studied

Lemma 2

The characteristic polynomials Pj(x) of Aj satisfy the following recur-rence relation:

(i) P1(x) = x− a1;

(ii) Pj(x) = (x− aj)Pj−1(x)− b2jv(j)Qj(x), j = 2, 3, . . . , n.

where Qj(x) denote the characteristic polynomial of the principal sub-matrix of Aj−1 obtained by deleting the row and the column indexed byv(j). As a convention, Q2(x) = 1.

Problem studied

Lemma 3

(Cauchy’s interlacing theorem) Let λ1 ≤ λ2 ≤ . . . ≤ λn be the eigenval-ues of an n× n real symmetric matrix A, and µ1 ≤ µ2 ≤ . . . ≤ µn−1 bethe eigenvalues of an (n− 1)× (n− 1) principal submatrix B of A, then

λ1 ≤ µ1 ≤ λ2 ≤ µ2 ≤ · · · ≤ λn−1 ≤ µn−1 ≤ λn.

Lemma 4

Let P (x) be a monic polynomial of degree n with all real zeros and λmin

and λmax be the smallest and largest zeros of P , respectively.

(i) If µ < λmin, then (−1)nP (µ) > 0.

(ii) If µ > λmax, then P (µ) > 0.

(iii) If P (µ) < 0, then µ < λmax.

Problem studied

Theorem 1

Let T be a tree labeled such that the adjacency matrix A(T ) is highlyacyclic. Then, the IEPT has a solution if and only if

αn < αn−1 < · · · < α2 < α1 = β1 < β2 < · · · < βn−1 < βn.

Further, in that case, there is a unique solution A ∈ S(T ) with positiveoff-diagonal entries. Any other solution differs from A only by signs ofsome off-diagonal entries. The solution is given by

aj =αjPj−1(αj)Qj(βj)− βjPj−1(βj)Qj(αj)

b2jv(j) =(βj − αj)Pj−1(αj)Pj−1(βj)

whereDj = Pj−1(αj)Qj(βj)− Pj−1(βj)Qj(αj).

Problem studied

The actual entries of the required matrix can be computed with SCILAB(or any other math software) by writing a computer program and feedingthe eigen data and plugging in the adjacency matrix as inputs. Theresults and solutions which appeared in the papers [5–9] are just specialcases of our result.

Numerical Example

Consider the generalized star T with 9 vertices shown in the fig-ure below. This tree was considered in [7] with the eigen data−60,−13,−8.83,−7.43,−2.7, 0.23, 2, 3.6, 4, 5.3, 10, 11.43, 12, 14.5, 15.64,21, 45 . The authors had labeled it as

Figure: A labelling of GS(3,2,3)

Numerical Example

Solution

The function v takes values v(2) = v(5) = v(7) = 1, v(3) = 2, v(4) =3, v(6) = 5, v(8) = 7, v(9) = 8. By our program we obtain the solutionas

4.00000 0.72111 0 0 7.24597 0 7.24283 0 00.72111 4.90000 3.71950 0 0 0 0 0 0

0 3.71950 7.24042 3.88790 0 0 0 0 00 0 3.88790 4.04923 0 0 0 0 0

7.24597 0 0 0 5.24265 7.79528 0 0 00 0 0 0 7.79528 0.10873 0 0 0

7.24283 0 0 0 0 0 0.58485 14.33160 00 0 0 0 0 0 14.33160 8.43147 47.351090 0 0 0 0 0 0 47.35109 −25.50266

which is the same as in [7].

Numerical Example

A different labelling

We relabel the same tree as

Figure: A labelling of GS(3,2,3)

Numerical Example

New solution

The function v takes values v(2) = v(4) = v(7) = 1, v(3) = 2, v(5) =3, v(6) = 4, v(8) = 7, v(9) = 8. By our program we obtain the solutionas

4.00000 0.72111 0 5.18175 0 0 8.70534 0 00.72111 4.90000 3.71950 0 0 0 0 0 0

0 3.71950 7.24042 0 5.47276 0 0 0 05.18175 0 0 7.73611 0 8.31937 0 0 0

0 0 5.47276 0 1.00253 0 0 0 00 0 0 8.31937 0 −2.02590 0 0 0

8.70534 0 0 0 0 0 −0.00739 13.41408 00 0 0 0 0 0 13.41408 9.55761 47.256670 0 0 0 0 0 0 47.25667 −26.41688

Numerical Example

Spectral constraints

The solution obtained satisfies the spectral constraints as seen below bycomputing the spectra of each leading principal submatrix.

σ(A9) = {- 60,−9.20564, −5.96072, −2.69053, 3.85537, 7.67418, 11.94503, 15.36881, 45}

σ(A8) = {- 13,−7.01439, −2.69380, 3.53984, 4.52978, 11.94470, 14.09725, 21}

σ(A7) = {- 8.83,−5.54438, −2.68935, 3.86974, 8.45463, 11.94512.15.64}

σ(A6) = {- 7.43,−2.69630, 2.43860, 4.09610, 11.94476, 14.5}

σ(A5) = {- 2.7, 0.29007, 3.94664, 11.34234, 12}

σ(A4) = {0.23, 2.27603, 9.94049, 11.43}

σ(A3) = {2, 4.14042, 10}

σ(A2) = {3.6, 5.3}

σ(A1) = {4}

Dominant solutions

Dominant Solutions

We refer the solution A to the IEPT that has positive off-diagonal entriesas the dominant solution. For each highly acyclic labelling of T , wehave a dominant solution of the corresponding IEPT. In this section,we discuss the number of dominant solutions that can be obtained fromdifferent highly acyclic labellings of T .

Dominant solutions

Dominant Solutions

We refer the solution A to the IEPT that has positive off-diagonal entriesas the dominant solution. For each highly acyclic labelling of T , wehave a dominant solution of the corresponding IEPT. In this section,we discuss the number of dominant solutions that can be obtained fromdifferent highly acyclic labellings of T .

For T ∈ T let us denote by ℓ(T ) the number of highly acyclic relabellingsof T . The dominant solution A obtained for a highly acyclic relabellingT0 of T is completely determined by the adjacency matrix of T0. Forany two highly acyclic relabellings of T , the dominant solutions coincideif and only if the corresponding adjacency matrices are identical. So,the number of highly acyclic relabellings of T giving rise to the samedominant solution A is

∣Aut(T )∣

∣, the number of automorphisms of T .Consequently, the number of dominant solutions obtained from different

labellings of T isℓ(T )

∣Aut(T )∣

Dominant solutions

We need to determine the number of ways of relabelling the vertices{1, . . . , n} of T such that for each j = 1, 2, . . . , n, the subgraph inducedby {1, 2, . . . , j} is a tree. Suppose T ′ is the resulting tree through such arelabelling. Note that n is pendent in the tree T ′, n−1 is pendent in thetree T ′ − n, and in general, j is pendent in the tree T ′ − {j + 1, . . . , n}.This observation facilitates a way for counting the number of highlyacyclic relabellings by reducing the problem to counting the same fortrees with lesser number of vertices.

Let P (T ) be the set of pendent vertices of T . For v ∈ P (T ) the numberof highly acyclic relabellings of T with v as the nth vertex is ℓ(T − v).We obtain the following recurrence relation for ℓ(T ).

ℓ(T ) =∑

v∈P (T )

ℓ(T − v).

Dominant solutions

Path Pn

Consider the path Pn on n vertices. Clearly, ℓ(P1) = 1. For n ≥ 2, Pn

has two pendent vertices, and deletion of each of them from Pn resultsPn−1. Thus, by repeated application of the recurrence relation,

ℓ(Pn) = 2ℓ(Pn−1) = · · · = 2n−1ℓ(P1) = 2n−1.

1 2 3 n− 1 n

Figure: The path Pn

Since∣

∣Aut(Pn)∣

∣ = 2, the number of dominant solutions obtained fromthe highly acyclic relabellings of Pn is 2n−2.

Dominant solutions

Star Sn

n− 1

Figure: The star Sn

Let Sn be the star on n vertices, n ≥ 2. ℓ(S2) = ℓ(P2) = 2. There aren− 1 pendent vertices in Sn and deletion of each gives Sn−1. So,

ℓ(Sn) = (n−1) ℓ(Sn−1) = · · · = (n−1)(n−2) · · · 3 ·2 ·ℓ(S2) = 2 · (n−1)!.

Now, the permutations of the pendent vertices produce all automor-phisms of Sn. So,

∣Aut(Sn)∣

∣ = (n − 1)!. So, the number of dominantsolutions obtained from the highly acyclic relabellings of Sn is 2.

Dominant solutions

The broom (or comet) Bn,m

The broom Bn,m has n + m vertices of which m + 1 vertices, namely,1, n+1, n+2, . . . , n+m, are pendent vertices. Since Bn,1 is the path Pn+1

and B1,m is the star Sm+1, we have ℓ(Bn,1) = 2n and ℓ(B1,m) = 2 ·m!.

1 2 3 n− 1 n

n+m− 1

Figure: The broom Bn,m

For n,m ≥ 2, the deletion of the pendent vertex 1 from Bn,m producesBn−1,m, and the deletion of each of the pendent vertices n+1, . . . , n+m

produces Bn,m−1.We get the recurrence relation

ℓ(Bn,m) = ℓ(Bn−1,m) +m · ℓ(Bn,m−1).

Dominant solutions

By mathematical induction, we see that for n,m ≥ 2 we have

ℓ(Bn,m) = m!n∑

im−1=1

im−1∑

im−2=1

· · ·

Now, the permutations of the pendent vertices n + 1, . . . , n + m pro-duce all automorphisms of Bn,m. So,

∣Aut(Bn,m)∣

∣ = m!. Consequently,the number of dominant solutions obtained from the highly acyclic rela-bellings of Bn,m is given by

im−1=1

im−1∑

im−2=1

Conclusion

On going and future work

1 We have dealt with the case of extremal IEP for matrices whosegraph is unicyclic.

2 The extremal IEP for an arbitrary connected graph is under study.

3 Finding the number of dominant solutions of IEPT for anarbitrary tree can be looked further into.

Bibliography

[1] L. Hogben. Spectral graph theory and the inverse eigenvalueproblem for a graph. Electronic Journal of Linear Algebra, 14:12–31, 2005.

[2] J. Peng, X. Hu, and L. Zhang. Two inverse eigenvalue problems fora special kind of matrices. Linear Algebra and its Applications, 416(2):336 – 347, 2006. ISSN 0024-3795.

[3] H. Pickmann, J. Egana, and R. L. Soto. Extremal inverseeigenvalue problem for bordered diagonal matrices. Linear Algebra

and its Applications, 427(2):256–271, 2007.

[4] H. Pickmann, J. C. Egana, and R. L. Soto. Two inverseeigenproblems for symmetric doubly arrow matrices. ElectronicJournal of Linear Algebra, 18:700–718, 2009.

[5] D. Sharma and M. Sen. Inverse eigenvalue problems for two specialacyclic matrices. Mathematics, 4(1):12, 2016.

Bibliography

[6] D. Sharma and M. Sen. Inverse eigenvalue problems for acyclicmatrices whose graph is a dense centipede. Special Matrices, 6(1):77–92, 2018. doi: https://doi.org/10.1515/spma-2018-0008.

[7] M. Heydari, S. A. S. Fazeli, and S. M. Karbassi. On the inverseeigenvalue problem for a special kind of acyclic matrices.Applications of Mathematics, 64(3):351–366, 2019. doi:https://doi.org/10.21136/AM.2019.0242-18.

[8] M. B. Zarch, S. A. S. Fazeli, and S. M. Karbassi. Inverse eigenvalueproblem for matrices whose graph is a banana tree. Journal ofAlgorithms and Computation, 50(2):89–101, 2018.

[9] M. Babaei Zarch and S. A. Shahzadeh Fazeli. Inverse eigenvalueproblem for a kind of acyclic matrices. Iranian Journal of Science

and Technology, Transactions A: Science, First Online 03 July2019:1–9, 2019. doi: https://doi.org/10.1007/s40995-019-00737-x.

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